Optimal. Leaf size=28 \[ \frac{\sqrt{-\alpha ^2+2 e r^2-\epsilon ^2}}{2 e} \]
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Rubi [A] time = 0.0046793, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {261} \[ \frac{\sqrt{-\alpha ^2+2 e r^2-\epsilon ^2}}{2 e} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{r}{\sqrt{-\alpha ^2-\epsilon ^2+2 e r^2}} \, dr &=\frac{\sqrt{-\alpha ^2-\epsilon ^2+2 e r^2}}{2 e}\\ \end{align*}
Mathematica [A] time = 0.0053698, size = 28, normalized size = 1. \[ \frac{\sqrt{-\alpha ^2+2 e r^2-\epsilon ^2}}{2 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.9 \begin{align*}{\frac{1}{2\,e}\sqrt{2\,e{r}^{2}-{\alpha }^{2}-{\epsilon }^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949937, size = 32, normalized size = 1.14 \begin{align*} \frac{\sqrt{2 \, e r^{2} - \alpha ^{2} - \epsilon ^{2}}}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68034, size = 58, normalized size = 2.07 \begin{align*} \frac{\sqrt{2 \, e r^{2} - \alpha ^{2} - \epsilon ^{2}}}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.543385, size = 36, normalized size = 1.29 \begin{align*} \begin{cases} \frac{\sqrt{- \alpha ^{2} + 2 e r^{2} - \epsilon ^{2}}}{2 e} & \text{for}\: e \neq 0 \\\frac{r^{2}}{2 \sqrt{- \alpha ^{2} - \epsilon ^{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08926, size = 22, normalized size = 0.79 \begin{align*} 0.183939720586 \, \sqrt{-\alpha ^{2} + 5.43656365692 \, r^{2} - 1 \times 10^{-24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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